Prédiction spatiale en présence d'erreurs substitutives et modélisation par drap stable

In geostatistics, the presence of outlying data is more the rule than the exception. Moreover, the statistical analysis of data contaminated by outliers requires caution, particularly when a spatial dependence exists. In order to take into account these possible outliers during the adjustment of the spatial process, a new modeling tool, called the substitutive errors model, is proposed. The optimal prediction in the least squares sense is derived and its properties are studied. Because of its complexity, this estimator needs in practice to be numerically approximated. An automated algorithm is proposed in this thesis. This method is based on an ordering of the observations with respect to the specified spatial process of interest, with the values least in agreement being included towards the end of the ordering. It proves to be useful in case of masked multiple outliers or nonstationary clusters. Simulations are carried out to illustrate its performances and to compare it to other forecasts, robust or not. An application to real data is provided as an illustration of its practical usefulness. The second part of this work also deals with the presence of spatial heterogeneity. One could say that the proposed model offers a characterization of this heterogeneity rather than estimating the locations and sizes of outliers. It is based on the theory of bidimensional α-stable motion. This represents a generalization of the unidimensional Brownian motion. In particular, the stability parameter α can be seen as a measure of the distance between the observations and the hypothesis of a Gaussian distribution. A method of estimation for the parameters of such a process is presented, based on a numerical constrained optimization of the likelihood. Its performances are illustrated by means of simulations. An application ends this second part.

A Quasi-Likelihood Approach to Zero-Inflated Spatial Count Data

Anthea Monod

The increased accessibility of data that are geographically referenced and correlated increases the demand for techniques of spatial data analysis. The subset of such data comprised of discrete counts exhibit particular difficulties and the challenges further increase when a large proportion (typically 50% or more) of the counts are zero-valued. Such scenarios arise in many applications in numerous fields of research and it is often desirable to infer on subtleties of the process, despite the lack of substantive information obscuring the underlying stochastic mechanism generating the data. An ecological example provides the impetus for the research in this thesis: when observations for a species are recorded over a spatial region, and many of the counts are zero-valued, are the abundant zeros due to bad luck, or are aspects of the region making it unsuitable for the survival of the species? In the framework of generalized linear models, we first develop a zero-inflated Poisson generalized linear regression model, which explains the variability of the responses given a set of measured covariates, and additionally allows for the distinction of two kinds of zeros: sampling ("bad luck" zeros), and structural (zeros that provide insight into the data-generating process). We then adapt this model to the spatial setting by incorporating dependence within the model via a general, leniently-defined quasi-likelihood strategy, which provides consistent, efficient and asymptotically normal estimators, even under erroneous assumptions of the covariance structure. In addition to this advantage of robustness to dependence misspecification, our quasi-likelihood model overcomes the need for the complete specification of a probability model, thus rendering it very general and relevant to many settings. To complement the developed regression model, we further propose methods for the simulation of zero-inflated spatial stochastic processes. This is done by deconstructing the entire process into a mixed, marked spatial point process: we augment existing algorithms for the simulation of spatial marked point processes to comprise a stochastic mechanism to generate zero-abundant marks (counts) at each location. We propose several such mechanisms, and consider interaction and dependence processes for random locations as well as over a lattice.

EPFL2012

Un modèle factoriel dynamique pour séries temporelles

Andrei Zenide

This work deals with factorial models for multiple time series. Its core content puts it at the interface between statistics and finance. After a brief description of the historical link between the two sciences, it reviews the literature on factorial models that are close to the model introduced in this work called The dynamical factor analysis model for time series. This model makes the hypothesis that the observed time series are influenced by a common factor, difficult to define and impossible to measure. No a priori structure is put on the factor, at each point time the value of the factor is considered as a new parameter that has to be estimated. As a consequence of this fact, the number of parameters is large and it is not possible to provide the usual asimptotic properties of the estimations by letting the number of periods tend to infi- nity. Asymptotic results in our context have been obtained by increasing the number of time series. The model makes the hypothesis that there is a linear dependence between the time series and the factor with coefficients, that are not constant over time, but rather follow a smooth random walk. This mean that the linear structure of the model is evolving slowly from a period to another. All the information which is not contained in the factor and the coefficients is considered as white noise. Using the normal distribution makes the estimation more easy and opens a toolbox of statistical methods that have been developed for this kind of data. The model is a part of the family of state space models and the Kalman filter is an essential ingredient of our estimator. The effort was concentrated on the elaboration of the structure of the model. Its complexity was constrained by the difficulty of estimation. The final shape of the model does not allow an analytical solution of the optimization problem introduced by the maximum likelihood estimation of the parameters. Numerical solutions have been found and compared with the parameters for simulated data. Some others models have been developed as simpler versions of the dynamical factor model for time series. The case where the factor can be observed has been studied and a new method for the estimation has been provided and compared with the existing methods. A second study considers a latent factor model without noise. Two methods for the estimation of the factor have been provided. The last chapter contains a detailed description of the main statistical tools used during this work. The links with the previous chapters are highlighted followed by comments.

EPFL2004

Contributions to Likelihood-Based Modelling of Extreme Values

Léo Raymond-Belzile

xtreme value analysis is concerned with the modelling of extreme events such as floods and heatwaves, which can have large impacts. Statistical modelling can be useful to better assess risks even if, due to scarcity of measurements, there is inherently very large residual uncertainty in any analysis. Driven by the increase in environmental databases, spatial modelling of extremes has expanded rapidly in the last decade. This thesis presents contributions to such analysis. The first chapter is about likelihood-based inference in the univariate setting and investigates the use of bias-correction and higher-order asymptotic methods for extremes, highlighting through examples and illustrations the unique challenge posed by data scarcity. We focus on parametric modelling of extreme values, which relies on limiting distributional results and for which, as a result, uncertainty quantification is complicated. We find that, in certain cases, small-sample asymptotic methods can give improved inference by reducing the error rate of confidence intervals. Two data illustrations, linked to assessment of the frequency of extreme rainfall episodes in Venezuela and the analysis of survival of supercentenarians, illustrate the methods developed. In the second chapter, we review the major methods for the analysis of spatial extremes models. We highlight the similarities and provide a thorough literature review along with novel simulation algorithms. The methods described therein are made available through a statistical software package. The last chapter focuses on estimation for a Bayesian hierarchical model derived from a multivariate generalized Pareto process. We review approaches for the estimation of censored components in models derived from (log)-elliptical distributions, paying particular attention to the estimation of a high-dimensional Gaussian distribution function via Monte Carlo methods. The impacts of model misspecification and of censoring are explored through extensive simulations and we conclude with a case study of rainfall extremes in Eastern Switzerland.

EPFL2019